[Paper Review] Lyapunov and Sylvester equations: A quadrature framework

Christian Bertram, Heike Faßbender|arXiv (Cornell University)|Mar 13, 2019

Matrix Theory and Algorithms31 references4 citations

TL;DR

This paper introduces a novel quadrature framework for solving large-scale Lyapunov and Sylvester equations using structure-preserving Runge-Kutta methods applied to associated ODE systems. By leveraging low-rank residual preservation through diagonally implicit Runge-Kutta (DIRK) schemes, the method achieves equivalence to the alternating direction implicit (ADI) iteration, enabling efficient low-rank approximations with a realification strategy for complex arithmetic.

ABSTRACT

This paper introduces a novel framework for the solution of (large-scale) Lyapunov and Sylvester equations derived from numerical integration methods. Suitable systems of ordinary differential equations are introduced. Low-rank approximations of their solutions are produced by Runge-Kutta methods. Appropriate Runge-Kutta methods are identified following the idea of geometric numerical integration to preserve a geometric property, namely a low rank residual. For both types of equations we prove the equivalence of one particular instance of the resulting algorithm to the well known ADI iteration. As the general approach suggested here leads to complex valued computation even for real problems, we present a general realification approach based on similarity transformation.

Motivation & Objective

• To develop a new numerical framework for solving large-scale Lyapunov and Sylvester equations using numerical integration of ODEs.
• To preserve the low-rank structure of the residual throughout iterative solution steps via geometric numerical integration principles.
• To establish equivalence between the proposed Runge-Kutta-based algorithm and the classical ADI iteration for both Lyapunov and Sylvester equations.
• To provide a realification technique based on similarity transformations to handle complex-valued computations in real problems.

Proposed method

• The solution of the Lyapunov equation is reformulated as the integral of a time-dependent Gramian, which is interpreted as the solution of a system of ODEs.
• A Runge-Kutta method is applied to the ODE system to generate low-rank approximations of the solution, with stability functions derived from Butcher tableaus.
• The method is constrained to preserve the low-rank property of the initial residual by requiring that the stability function satisfies a specific functional identity.
• For the Sylvester equation, a similar ODE system is derived, and the same Runge-Kutta approach is applied, yielding a low-rank iterative scheme.
• The framework identifies DIRK methods as those preserving the low-rank residual, leading to equivalence with the ADI iteration.
• A realification strategy via similarity transformation is introduced to convert complex-valued Runge-Kutta iterations into real arithmetic for real problems.

Experimental results

Research questions

• RQ1 Can a quadrature framework based on ODE integration be used to derive low-rank iterative solvers for Lyapunov and Sylvester equations?
• RQ2 What class of Runge-Kutta methods preserves the low-rank structure of the residual in the solution process?
• RQ3 Is the resulting algorithm equivalent to the well-known ADI iteration for both Lyapunov and Sylvester equations?
• RQ4 How can complex-valued computations arising in the method be transformed into real arithmetic for practical implementation?
• RQ5 What is the connection between the stability function of the Runge-Kutta method and the rational Krylov subspace framework?

Key findings

• The proposed Runge-Kutta-based algorithm for the Lyapunov equation is mathematically equivalent to the ADI iteration when using DIRK methods.
• The method preserves the low-rank structure of the residual throughout iterations by construction, ensuring computational efficiency.
• For the Sylvester equation, the same framework yields an algorithm equivalent to the ADI variant for Sylvester equations, as shown in Algorithm 4.
• The stability function of the Runge-Kutta method must satisfy a specific functional identity to preserve low-rank residuals, which characterizes the class of valid methods.
• The realification approach via similarity transformation allows the use of complex arithmetic methods in real-valued problems without loss of accuracy or efficiency.
• The framework provides a unified derivation of ADI-type methods from ODE integration, linking geometric numerical integration with rational Krylov and moment-matching approaches.